Question

Give the numerical value of these expressions.
a) P(X > 4) if X~Binomial(8, 0.65).
b) P(5 < X ≤ 7) if X~Binomial(8, 0.65).
c) P(X > 3) if X~Poisson(7.8)

Answer 1

X ~ Binomial (n,p)

P(X) = ⁿC_x p^x (1 - p)^n-x

a)

P( X > 4) = P (X >= 5)

= P( X = 5) + P( X = 6) + P( X = 7) + P( X = 8)

= ⁸C₅ 0.65⁵ 0.35³ + ⁸C₆ 0.65⁶ 0.35² + ⁸C₇ 0.65⁷ 0.35 + ⁸C₈ 0.65⁸ 0.35⁰ ]

= 0.7064

b)

P(5 < X <= 7) = P( X = 6) + P( X = 7)

= ⁸C₆ 0.65⁶ 0.35² + ⁸C₇ 0.65⁷ 0.35

= 0.3959

c)

X ~ Poisson()

Where = 7.8

P(X) = e^-X / X!

So,

P( X > 3) = 1 - P( X <= 3)

= 1 - [ P (X = 0) + P (X = 1) + P (X = 2) + P (X = 3) ]

= 1 - [ e^-7.8 + e^-7.8 * 7.8 + e^-7.8​​​​​​​ * 7.8² / 2! + e^-7.8​​​​​​​ * 7.8³ / 3! ]

= 0.9515